Theory of Events, Space and Time - · PDF fileTheory of Events, Space and Time a neoclassical...

Theory of Events Space and Timea neoclassical relativity

Author Corrado La Posta Viterbo Italy August 3rd 2017

Abstract

The Theory of Relativity [1] conceived in 1905 by A Einstein and now universally applied involves some theoretical problems as well as some difficulties in theinterpretation of experimental facts Theoretical issues have long been debated [24] but have not found a sufficiently good explanation indeed many interpretations [324] appearforceful and prejudicial The many paradoxes expressed and debated and never clarified are still unresolved The consequences of adopting a radical point of view such as the relativistic one also appear in the Quantum Field Theory where the Lorentz-covariance formulation of the theory results with careful analysis the cause of many of the problems[54 78]

The experimental point of view contrary to what is commonly stated does not undoubtedly support the theory actually several results [542 78] appear of difficult interpretation and others [6 75] even in sharp contrast to the theory itself [23 76]

The analysis of these inconsistencies leads to the assumption that the theory is spoiled by its foundations that is the postulates from defective origins that can not be eliminated [40 77] An alternative hypothesis based on different premises is therefore elaborated in this paper to reach a number of conclusions [81]

Such a different theoretical context recovers certain concepts of classical mechanical and electrodynamic physics which are however extended in a relativistic sense Modifications are introduced in current kinematic dynamic electrodynamic and Quantum field theory The theory thus developed is devoid of paradoxical aspects adhering to experimental facts and free of divergence problems superluminary motions are especially possible Electrodynamic equations are extended into a new invariant formcompatible with Newtonian mechanics Lorentzs force law is rewritten as a force variable with speed while the mass remains constant A generalization of the equation of the waves is introduced from which the equations of the superluminal motion and the equations of the material waves (Schrodinger and Klein-Gordon) can be obtained

1

Index

1) Symbols

2) Special Relativity

3) Ether theory

31 - Michelson amp Morley

32 - Sagnac amp Michelson-Gale

33 ndash Cosmic Background Radiation

34 ndash More tests

4) Euclidean Space-time

41 ndash Length contraction

42 ndash Time dilation

43 ndash The γ factor

5) Electrodynamics

51 ndash Maxwell equations in invariant form

52 ndash Electromagnetic waves

53 ndash Lorentz Force

6) Relativistic effects

61 - Doppler effect and Ives amp Stilwell experiment

62 ndash Compton effect

64 - Fine structure of the Atom

66 - Hafele-Keating experiment

7) Schrodinger equation

2

1) Symbols

MM - Michelson ndash Morley Experiment

QM - quantum mechanics

GTR - general theory of relativity

STR - special theory of relativity

AF - absolute frame

IF - inertial frame

NIF - non inertial frame

OI - observer inertial

ET - ether theory

TEST - Theory of Events Space and Time

GT - Galileian transformations

LT - Lorentz transformations

QFT - quantum field theory

3

12) The Special Theory of Relativity

Relativity [1] was born as a response to the problems of electrodynamics of the nineteenth century in particular the difficult interpretation of the experiments of Fizeau [7] Bradley [8] Michelson-Morley [9] as well as theoretical problems raised by ether theories It also develops on the basis of A Einsteins philosophical reflections on the concepts of space and time the concept of measurement and the main electrodynamic phenomena Several attempts at theoretical framing of the facts had been tempted by various physicists [1114] first of all AHLorentz [12] but only with Einstein has the emergence of a unitary theory able to explain coherently the various phenomena The various theoretical arrangements while providing similar results differed in some places deeply Lorentzs theory in particular explicitly and positively laid down a particular substrate the ether the seat of electromagnetic phenomena which is quite absent in the Einsteinian formulation

The revolution brought by Einstein to the physics of the nineteenth century is based on two postulates the first of which affirms the equivalence valid in principle of all inertial observers while the latter affirms the constancy of the speed of light always with reference to inertial observers The Newtonian concepts of absolute space and time are completely abolished while the theory is developed in a covariant form [22] that is valid for all observers

The principle of the constancy of light velocity at first sight somewhat paradoxical is the basis of the theoretical difficulties that relativity has encountered over the years including paradoxical situations [13 75] asymmetries of difficult understanding presence of infinite in the theory and more that have stimulated various reflections [24 78] and heavy criticism [1557 77]

The initial successes in the explanation of experimental facts over the years have been supercede by doubts both for the observation of phenomena [559] which are framed with difficulties in the relativistic context and by the presence of other [6 58] thatseem openly contradictory to theory

The abolition of the concept of ether on the other hand turned out to be somehowimprovised [1617] or even as deleterious as the developments in QFT [18] or GTR [19] On several occasions [20] [21] the concept of ether has been re-evaluated and deepened with remarkable results in various fields of physics [35] Many of the attempts made however are an unfortunate compromise that tries to save a little bit of everything failingto satisfy anyone fully sometimes leading to the introduction of obscure or extremely questionable concepts [2425 76]

The review of the concepts of space and time proposed by Einstein appears to the eyes of modern critics [26] spoiled by prejudices as well as the apparent cinematic phenomena of contraction of the lengths and dilation of times which sometimes seem unconvincing and sometimes even surprisingly unlikely [27] The disagreements with theQM appear to be related to the well-known problems of non-local interactions [68697071]

The adoption of the Lorentz-invariance concept has led to major theoretical problems both in QFT and RG

In the first case there are well-known problems of the presence of infinites and problems of renormalization [1854] while in the latter there is difficulty in the

4

elaboration of cosmological models [28] which is moreover obviated by abolishing the pseudo- Euclidean minkovskian metric [1922]

The exceptional development of STR [29] and its application in almost all fields of physics have however curbed the enthusiasm of critics since abandoning or replacing it with new concepts inevitably leads to a titanic reconstruction work of almost all physics If however it is understood the conceptual and practical difficulties the theory has created over time can also be understood as such unwise task is almost inevitable and indeed necessary

23) Ether theories

The first attempts for a mechanical or hydrodynamic description of the background space date back to the newtonian era and over the years the various models and different interpretations of electromagnetic phenomena have multiplied The most promising attempt was the one by AHLorentzs relating to an immobile ether tied to absolute space This conception was however criticized [3062] because of the introduction of some auxiliary hypotheses including that of contraction which seemed unlikely if not opportunisticAnd was then overcome by the limited relativity that avoided the use of additional hypotheses based only on two postulates

The theoretical and experimental problems encountered in STR have led several authors to re-evaluate the concept of ether both in relativistic form [3132] andor in a new lorentzian way [3334] The theoretical debate has also been extensively deepened both with STR and ET criticism and with new theoretical proposals The concept of etherwas also implicitly adopted in quantum mechanics [63] gravitation theory [19] field theory and various other fields [61]

In QM and QFT in particular a complex dynamic quantum vacuum structure [64] is evident which can obviously be taken as the basis for a modern conception of the etherconcept

In the simple and yet complete elaboration of Lorentz the ether is the seat of electromagnetic phenomena in particular of the light phenomena and constitutes the propagation medium of em waves Such waves propagate at constant speed with respect to that medium which therefore constitutes a privileged reference regardless of the state of motion of the source The velocity of the waves is constant only in a SR at rest with respect to the ether and not for a uniform moving observer (OI) The difference is evidentfrom the relativistic point of view but the results obtained are perfectly consistent with the premise and the concepts of classical physics

5

In fact letrsquos have a spherical wave propagating in the bottom space with velocity c it describes a radius sphere r = ct and equation

c2t 2=x2 (31)

In a reference in motion in the x direction the wave seems propagating in a modified shape in fact transforming according to galileo

x =xminusv t t=t x=x +v t t=t (32)

(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)

( c2minusv2 iquest t 2minus2 x vt minusx 2=0

And the propagation speed results for a motion along one of the axes

u = crsquo = minus2 vplusmnradic4 v2+4c2minus4v2

2

(34)

that is

c = c-v c= - (c+v)

depending on whether the observer moves away or approaches the sourceThe speed of light c is connected to the two quantities ε0 and μ0 from Maxwells relation

c = 1

radic ε 0 μ0(35)

Which is here interpreted in a modeling sense with ε0 and μ0 characteristic magnitudes ofthe ether The concept of ether is here to coincide with that of the absolute space of Newtonian physics coupled with particular physical properties The equations of Newtonian mechanics apply to all OIs ie those in straight-run motion with respect to the privileged reference Maxwellian electrodynamic equations are valid only in the rest frame of the ether while for OIs need to be appropriately modified (sect51) This explains the constancy of the speed of light holding only in absolute reference systems being the velocity of propagation of a wave and as such dependent on the characteristics of the medium

The mechanical description of the ether had already been tried in the past centuries [30] with different models especially elastic These descriptions encountered various problems and were therefore abandoned maybe prematurely

6

31) Michelson and Morley Experiment

AAMichelson and EWMorleys interferometric experimentation is generally regarded as crucial in the development of modern scientific thinking and as a mile stone in the crisis of classical physics A model interpretation of the effect was given by AHLorentz almost immediately linked to his theory of ether and was based on a dynamic contraction effect (the famous Lorentz-Fitzgerald contraction) a real effect due to dynamic phenomena associated with the structure of the ether

The cancellation of the effect of the second order is far from fortuitous or surprising but perfectly consistent in the new description of the electrodynamic phenomena

Therefore turning the apparatus slowly or quickly can not in any case be seen as a second-order effect The possibility of a residual effect of the third order remains open Even in the long debated [42] Miller results there is a residual effect of the same order of magnitude compatible with current knowledge of the state of the Earths motion [72]The subsequent repetitions of the interferometric experiments also show the presence of the effect remaining silent though and still without explanation

It has been said [15] that the total effect may depend on the details of the experiment where there are multiple reflections slightly unequal arms reflection angles not exactly equal and not exaggerated equal to π 2 and π 4 adjustment Mirrors and other details A complete calculation of the optical paths in the real apparatus however shows that this conjecture is to be rejected and that the effects considered are erased exactly at least to the the second order

32) Sagnac and Michelson-Gale Experiments

A variant of the MM experiment is based on the use of a rotating interferometer first made by MGSagnac [43] and in a large-scale variant by Michelson himself along with HGGale [44] The results obtained were immediately and clearly interpreted in full agreement with the hypothesis of the stationary ether both by Sagnac and Michelson while the relativistic explanation appeared immediately more difficult The first attempts [45] of explanation have followed many controversies [465560] between relativists and opponents and the heated debate still rages todayIn view of the new theory of the ether the effect seems completely natural if not trivial Sagnac calculation provides

∆ t = (t1 ndash t2 ) = 2πr(c+ ωr)+2πr(c- ωr)(312)

∆ t ~ 4ωAc2 (A=path area ω=angular velocity)

Note in particular how Langevins (relativistic) computation [45] does not require the use of STR but GTR and hides the (ab)use of an absolute reference and a Galilean transformation implicitly contradicting the Principle of Relativity The calculation is compatible with what is developed in the follow-up of this theory in an euclidean hyperspace (sect41)

7

In the case of a rectangular path such as that of the Michelson-Gale experiment as well as for other possible paths the result obtained [44] is always given by (312) which is therefore generally validIn the rotating (non-inertial) system the calculation is not much more complex and usingthe appropriate transformations (sect41) the same final result is obtained The speed of light is obviously not constant in the Einsteinian sense but only in a narrower sense ie relative to the reference frame of the propagation medium of the em wavesA generalization of (312) is possible to include other earthly motions (revolution around the Sun rotation of the Galaxy etc )

∆ t ~ 4(ωrot + ωriv + ωgalactic + hellip)Ac2 (313)

but as we see immediately the additional terms are negligible compared to the one due to the Earths rotation as the result depends on the angular velocity and not the absolute one The situation is opposed to that of MM type experiments where the dominant term is due to the global motion of the Earth relative to the CBR see also sect34

33) Cosmic Background Radiation

A crucial aspect in the analysis of theories of ether lies in the study of the fundamental cosmic radiation The radiation appears from the Earth slightly anisotropic with a simple sinusoidal pattern The explanation of the effect is surprisingly trivial it is due to the motion of our planet with respect to the ether with a speed of about 370 kms [6] Such an explanation is however inadmissible in the relativistic context where there is no privileged reference and often attempts at reconciliation between theory and experiment overcome ridicule [1525]In the new theoretical context the very presence of the background radiation which can be understood as a thermal effect of the ether is quite natural

34) More tests

The original MM type interferometric experiments with incoherent light have been replaced in modern times with the use of Maser [48] or Laser [49] devices Accuracy and sensitivity are steadily increasing as well as the sophistication of the equipment Even in these cases however there are residual effects of the type referred to above (sect31) whichcan be explained by the dynamic effects of the ether

Of great importance in this context are the works of HAMunera who in addition to having shown remarkable subtleties from a theoretical point of view [72] also realized a modern variant of the experiment with a non-rotating interferometer that gave a full effect The greatest care in detecting the causes of error and modern data analysis procedures allowed Munera to succeed where Michelson had failed [73]

8

34) Dynamics

One of the highlights of STR is the use of a four-dimensional space-time description due mainly to HMinkovsky [50] In this description a wide-ranging use of four-dimensional notation 4-vectors and tensors is made The use of a pseudo-euclidean metric is also essential

gμν = (1 0 0 00 minus1 0 00 0 minus1 00 0 0 minus1

) (41)

and the Lorentzs transformations are

x =γ (x+β ct ) t=γ ( t+βx c) (42)

Since the second postulated ie the constancy of c requires the invariance of DAlembertsequation

part2

c2 part t2 ϕ=∆ ϕ (43)

In TEST the constancy of velocity c is interpreted in a completely different way so Galileian transformation laws and the use of an euclidean metric tensor in 4-dimensional hyperspace are appropriate The wave equation is not invariant according to (43) but in an extended form Maxwells equations are also invariant in an extended formWith these considerations the four dimensional formalism can be maintained with minimal modificationsThree types of references will be used in what follows

- Absolute frames at rest with respect to the ether- Inertial frames in rectilinear motion with respect to the ether and relative to each other- Non-inertial references otherwise

The most noteworthy difference to the relativistic paradigm but also to Newtonian physics is the introduction of absolute references It is understood that references that areat rest in relation to Maxwellrsquos ether that do observe an isotropic CBR that coincide withthe Absolute Space of Newton or that coincide with the Einstein Cronotope in the GTR are one and the same These different definitions are hereby supposed to be equivalent at least for the time being by reserving to better define the meaning if new experimental observations allow or suggest some difference between the four above-mentioned realities or indicate the preference of a definition with respect to another

9

We anticipate that Maxwells equations in their simplest and traditional form are valid only in absolute references corrections are introduced in the other referencesIn an absolute reference system (AF) the metric tensor in empty space is given by (see also [74])

gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)

So the space-time interval results

ds2=gμν dxμ dxν (45)

The wave equation (43) is expressed by the wave tensor

Δμν = Δμν = (1 0 0 00 minus1 0 00 0 minus1 00 0 0 minus1

) (46)

So that

Δ μν partμ partνφ=0 (47)

And the speed and acceleration 4-vectors of a moving particle are defined as

uμ=(c v ) aμ=(0a) (48)

From which the 4-force (of Newton)

f μ=m aμ (49)

Note that in this reference the covariant components coincide with the contravariant onesMomentum and energy are given by

pμ=( Ec

m v ) (410)

pμ pμ

m=m c2 radic1+β2 (411)

and the usual limits hold β rarr0 E = m c2+ 12

m v2

β rarr infin E = mcv = pc

10

Turning to a moving system (IF) according to Galileo


translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)

x μ=Λ νμ xν (414)

The 4-velocity becomes

u μ=Λ νμ uν = (c u - v)

(415)

And the 4-force is invariant

f μ=dpμ

dt=(0 a) (416)

The derivative 4-vectors are

partμ=( partc part t

minus vc

partpart x

part

part x) partμ=( part

c part t

partc partt

+ vc

partpart x

) (417)

The wave tensor transforms according to

Δμν=(1minusβ2 minusβ 0 0minusβ minus1 0 00 0 minus1 00 0 0 minus1

) (418)

and yelds the generalized wave equation

((1minusβ2) part2

c2part t 2 minus2βpart

part xpart

cpart tminus part2

part x2)ϕ=0 (419)

To be compared to (33)

11

(419) is the generalization of (43) and still admits solutions in the form of waves of the type Φ (x + ut + k) Ψ (x-ut + h) with u given by

c (β plusmn 1) (420)

In accordance with what is stated in sect31 Specifically (419) is valid for a motion with a higher speed than c where there is propagation of superluminal waves The velocity c in particular in this exposition is no longer a theoretical limit nor is there any magnitudes that diverge when v tends to c

41) Length contraction

The explanation of the null effect in the experiment of MM recalls the phenomenon of contraction [65] however in the theory of ether this loses its hypotheticalaspect to become a dynamic phenomenon connected to electromagnetic interaction between electrons and protons We know today that matter is held together by electrodynamic forces and that they are responsible for the shapes and dimensions of the bodies The behavior of atomic systems is governed by the Schrodinger equation in which both dynamic terms and terms of electromagnetic interaction appear As mentioned however and as you will see (sect51) Maxwells equations are valid in their classical and simpler form only in Absolute Frames while in Inertial Frames they are invalid Therefore (small) corrections should be introduced in the Schrodinger equation in the LCAO MLV and in the rest of quantum formalism when referring to bodies in motion This is why (sect53) an anisotropic expression for Lorentzs force is acting on a moving body that involves anisotropy in the wave function In particular there is a crush in the direction of motionAs a result a spherical orbital crushes into a rotational ellipsoid of sortThe cancellation of the effects of course certainly dynamic loses its character of artificiality and looks completely natural with different effects having the same cause at its origin It is evident that the effect is neither apparent nor introduced ad hoc It does not even seem appropriate to insert it into the transformations of coordinates which remain simply galileian formulas (412) [53]

12

42) Time dilation

The dilationcontraction of time is seen here as a real (and not apparent) physical phenomenon but belonging to the measuring instrument not to the physical time In fact if you use an interferometric system or a Laser or any other device that uses electromagnetic waves the above effects are encountered For example a light-clock in an IF due to the anisotropic propagation phenomena of waves measures a time given by

t=lcm (424)

on the longitudinal arm where cm is the average speed of light

cm=c0 (1-β2 ) (longitudinal arm) (425)

Similar results are obtained for the transverse arm which coincides at least to the second order

43) The γ factor

The coefficient γ=radic1+β2 of (411) in what follows will sometimes go replacing the

one used in STR γ= 1

radic1minusβ2 assuming the same values (except for β) but with the

remarkable difference that the new one has no divergence problems and superluminal speeds become an absolutely common factor The factor γ=radic1+β2 together with a revised Lorentz Force formula (sect 53) allows to re-interpret many experiments or facts (average life or rather average path of muons synchrotron frequency etc ) without introducing any divergence in the equations

13

45) Electrodynamics

One of the fundamental requirements in STR is the invariance of the electrodynamic equations in the form given to them by JCMaxwell This need is purely theoretical and reflects the search for a symmetry that is imagined in nature itself Since these equations are not invariant to Galileos transformations they have led to the introduction of Lorentzs transformations However there is no guarantee that there are no other and different ways of describing the phenomena By developing Galilean formalism and its own invariance we will naturally be brought to different conclusions including the extension and generalization of the Maxwell Equations the generalization of the DAlembert equation the extension of quantum formalism All this in perfect consistency with the premises and in accordance with classical physics so there is no need for any changes as it happens in RR

In practice while in relativity it has been made to modify the mechanics to make it compatible with electrodynamics we will see that in the present scenario is the electrodynamics that is modified in a completely natural way to put it in agreement with the mechanics always guided by the same desire that of a unified description of physicalphenomena [52] [41]

51) Maxwell Equations in invariant form

Let starts with the 4-dimensional potential as usual

Aμ = (φ A)

From which the E and B fields are derived

E = minusnabla φminus partpartt

A B = nabla times A

Letrsquos compute the EM tensor

Fαβ = (0 minusEx minusEy minusEzEx 0 minusBz ByEy Bz 0 minusBxEz minusBy Bx 0

)(51)

The Lorentz Force yields

f μ = (0 a) (52)

In a moving inertial frame the EM tensor transforms as

14

Frsquoαβ = Λαμ Λβ

ν Fμν =

(0 minusEx minusEy minusEz

Ex 0 minusBz+βxEyminusβyEx By+ βxEzminusβzExEy BzminusβxEy+ βyEx 0 minusBx+ βyEzminusβzEyEz minusByminusβxEz+βzEx BxminusβyEz+ βzEy 0

) (53)

f rsquoμ =(0 a) (54)

with E = E B = B - β times Ec

the Lorentz Force (54) is generally anisotropic

part pα

partt=q

cF

α β uβ

(55)

The 4-potential becomes

(56) Arsquoμ = (φ A+φ β )

And the 4-current is

J α=iquest (ρ J+ρβ)(57)

Finally using formulas from chapter 4 ( partμ = (part

cpart t + β nabla nabla iquest etc hellip )

partμ J μ=0orexplicitly part ρcpart t

+ β nabla ρ + nabla J+ρβnabla = 0 (58)

Maxwell Equations

partμ Fμρ=1c

J ρ together with partμ Fωρ+partω Fρμ+partρ Fμω=0 (59)

15

through (55)-(58) become [41]

in explicit notation in the four classical fields (E B=μH D=εE H)

(510-12)

part Bpart t

+ (vnabla ) B=minus(nabla times E )+[(Bnabla )vminus(nabla v ) B]

part Dpart t

+(vnabla ) D+J=(nabla times H )+[( Dnabla ) vminus(nabla v ) D ]

nabla B=0 nabla D= ρ

The (510) contain v-dependent terms however they are perfectly invariant in the most general sense as their writing in tensorial notation showsTheir solution [41] shows exactly the type of predictable behavior in an inertial motion system with speed v with respect to the ether

552) Electromagnetic waves

The (510) in the case ρ = 0 J = 0 v = 0 represent the DAlembert equation for the electromagnetic field They are valid with respect to an AF and as in Chapter 4 they are transformed according to (418) by carrying an IF into a type (419) equation

For a slow motion system (β ltlt 1) terms in β and β2 can be overlooked and the DAlembert equation (43) is foundFor β = 1 that is for a system in motion at the speed of light two terms cancel and the equation becomes

part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)

From which it is possible among other things to derive the phenomenon of the optical bangFor superluminal velocities terms in β are dominant and with good approximation

partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)

which we will call the equation of superluminal wavesObviously itrsquos not that electromagnetic waves propagate at a speed different than c the speed of light is always the same in an AF but how a fast-paced observer sees these waves At no point is it forbidden to travel at speeds greater than c but it would seem (see

16

sect41 and sect53) that matter as we know it ceases to exist when approaching c precisely because of spatial contraction phenomena It can not be ruled out however that particularobjects (particles nuclei) held together by non-electromagnetic forces (nuclear gravitational) can travel superluminally The question of the propagation speed of non-electromagnetic interactions such as gravitational interaction [64] as well as that of the existence of tachyons pops up again 653) The Lorentz force

The (55) calculated for a motion reference with velocity v = βc with respect to the background space is for a particle with velocity u

part p α

partt=f α=q

cF

αβ u β

(0 a)=(0 minusEx minusEy minusEzEx 0 minusBz+βxEyminusβyEx By+ βxEzminusβzExEy BzminusβxEy+ βyEx 0 minusBx+ βyEzminusβzEyEz minusByminusβxEz+βzEx BxminusβyEz+ βzEy 0

)(c+β ∙u

ux

uy

uz

)that writing down the spatial components of f α yields

f =qc

(c E+u times Bminus(E ∙u) β) = m a (54)

A paradigm shift should be noted with respect to STR it is not the mass that increases with speed but the force (electrodynamic) decreasing with it see also [82]

as v rarr c one obtains f elrarr0

The β correction term goes multiplying the force not dividing the mass freeing the equations from the presence of infinites Bodies still canrsquot accelerate to the speed of light when pushed by electromagnetic forces alone

17


The theory of relativity envisions and explains a whole series of phenomena that are usually brought as evidence of the goodness of the theory and the agreement between theory and experimentHowever it is not ruled out that such phenomena can be framed in a different context andexplained in a different conceptionBelow is described the interpretation of some of the most important experimental facts onthe basis of this theoretical setting Effects can still be considered relativistic but obviously the terms meaning is different from the ordinary

61) Doppler effect and Ives and Stilwell experiment

The discovery of the relativistic formula of the Doppler effect including transverse effect is one of the results that AEinstein obtains immediately and has been tested for the first time by HEIves and GRStilwell [8]This effect is found in this conception as well as the agreement with the experimental facts First consider the case of a source of uniform straight motion with respect to the ether and a silent observer (AF)Let O and Orsquo be the AF points where two consecutive circular waves are emitted and indicate with

t1= time of emission of the first wave

t2=t1+T time of emission of the second wave

t1= time of arrival of the first wave

t2= time of arrival of the second wave

18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc

R=(Rcos-vT)2+r2sen2 (62)

t2=t1+T+Rc

a Taylor expantion gives

t2=t1+T+Rc(1-vTRcos+v22c2sen2)+O(vTR)3 (63)

T=t2-t1=t1+T+Rc(1-vTRcos+v22c2sen2)-t1-Rc (64)

=T(1-vccos+v2T2cRsen2)

Since by definition of wavelength RT=c

T=T(1-vccos+v22c2sen2) (65)

The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)

In the case of a source at rest and a moving observer the formulas differ but the process is essentially the sameObserving the radiation in two opposite directions as in the Ives and Stilwell experiments you get

= 1=(1-vc cos+v22c2sen2)

=+ 2=(1+vc cos+v22c2sen2)

m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19

62) Compton Scattering

The Compton effect is usually considered to be a relativistic effect but it can also be derived within the scope of this formulation Collissions are represented by the following diagram

Energy and momentum conservation yields

hoc=hccos+movcos (67)

0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which

(hc)2(o-cos)2=mov2cos2

and a few more computations

-2(hc)(-o)moc=(hmoc2)o(1-cos)

Introducing wavelength

-o=(hmoc)(1-cos) (68)

That is the usal formulationThe kinetic energy of the electron is

T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h

o(homoc2)(1-cos)(1+ho(1-cos)moc2) (610)

verified for the first time in the Bless experiment

64) The fine structure of the hydrogen atom

One of the classic and most persuasive applications of relativity theory is to calculate corrections to the structure of the spectral lines highlighting a fine structure The process uses perturbative methods by introducing corrections in the Hamiltonian systemIn this theory the method is equally applicable and although the kinetic energy expressionslightly differs from the usual one the energy-impulse relation is the same as in the relativistic case The corrective terms are also the sameLetrsquos consider a hydrogen atom with classic hamiltonian

H0=p122m1+p2

22m2+V(|r1-r2|) (611)

In the center of mass frame

H0=r2+V(r) con =m1m2(m1+m2)

The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)

From which a Taylor expansion gives

H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))

That approximatly equals

H=H0+H1 with H1=-(Pr48m3c2)

21

66) The Hafele-Keating Experiment

One of the most accurate examinations of the time dilation effect was performed by JCHafele and REKeating in 1971 [67] The explanation based on the neo-relativistictheory of ether does not differ in the aspects of calculation from the usual one except for the interpretation of the effect that must be considered due to the absolute motion of the watches and not the relative one (compare sect 42) The effect due to the Earths revolution and translational motion is in fact giving a minor contribution and only the effect of the rotation motion remains in addition to the motion of the watches of course The gravitational corrections used by the authors must also be considered in the new theory in fact as we will see in the theory of gravitation only small adjustments appear and the calculation of many effects results in similar values to those already known

Similar considerations can be made for the GPS localization system The effect is interpreted as absolute and not relative due to the motion of clocks with respect to the ether The effect is also apparent that is instrumental the time magnitude does not undergo any real dilation it is the drifting tool that does however in a correctable manneras it is predictable

Finally note how the Twins Paradox loses all its mystery in the present context Since motion is no longer relative but absolute it is always possible to determine which twin ages faster (only an apparent effect) or which clock slows down (real effect)

22

7) Schrodinger Wave equation

Equation (419) con be approximated for small speeds (βltlt1) and slowly varying fields

(part2

c2 part t2 ϕ=0 ) as

(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)

With the usual correspondences ( E rarri hpartpart t

Prarrminusihnabla ) and computing the mixed

term it provides the Material Wave Equation [64]

(ihpartpart t

minus h2

2m∆)ϕ=0

(72)

In this context the interpretation is the following taking also into account a subluminal group and superluminal phase velocity the Schrodinger equation represents the electron point of view on the wave equation that is how a moving material particle sees the quantum substrate (ether) to evolve

The point of view of the photon is instead that of DAlemberts equation (vg = vf)

For high speeds (βgt 1) from (419) it is obtained neglecting a term

( part2

c2part t 2minusβ2 part2

c2 partt 2 minuspart2

part x2)ϕ=0 (73)

that again ( E rarri hpartpart t

Prarrminusihnabla ) brings the Klein-Gordon equation

( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)

Dirac equation follows from (74) as usual

23

7 References

[1] AEinstein Zur Elektrodynamik bewegter Korper Annalen der Physik 17 891 1905[2] see CMoller trionfi e limiti della teoria di Einstein in Centenario di Einstein 1979[3] AAUngar Foundof PhysVol19No1113851989[4] LEssen relativitagrave scherzo o truffa Inediti Andromeda[5] vedi pes EJPost Reviews of Modern Physics vol39n2p475aprile 1967[6] vedi pes PLubin ed altri The Astrophysical Journal 298L1-L5Novembre 1985 or SMarinov General Relativity and Gravitation 12571980 References in JPWesley Foundations of Physics Vol16No88171966[7] AFizeau Comptes RendusXXXIIIp3491851[8] Ives H E Stilwell G R (1938) An experimental study of the rate of a moving atomic clock Journal of the Optical Society of America 28 (7) 215[9] AAMichelson EWMorley Philosophical Magazine S5Vol24N151p449Dec1887[10] WVoigt Gottinger Nachtrichten 411887[11] HPoincareacute Comptes Rendus 14015041905[12] AHLorentz The Theory of Electrons 1906 in Collected Papers[13] see WARodrigues ECDe Oliveira Physics Letters A vol140N94791989[14] see JLarmor The Scientific Writings of the Late george Francis Fitzgerald 1902[15] RMonti Seagreen n1 nov1984n2 feb1986n56 inverno 198788[16] AKAMaciel JTiomno Foundations of Physics Vol19No55211989 [17] GSparvieri GContreras Il Nuovo Cimento Vol91 N21431986[18] as an introduction to QFT IF see FMandl GShaw Quantum Field Theory[19] PDPesic Il Nuovo Cimento Vol108BN1011451993[20] GCannata Letere questo sconosciuto Universitagrave di Palermo 1981[21] FSelleri Zeitschrift fur Naturforschung 46a4951990 [22] invariance analisys in JNorton Found of Physics Vol19No1012151989[23] IF veda pes SMarinov Eppur IF muove 1979 e anche JCasares Roldan Refutacion de los fundamentos de la relatividad 1952[24] see GArcidiacono La relativitagrave dopo Einstein Di Renzo editore 1993 and also

24

ILicata La realtagrave virtuale Di Renzo editore 1991[25] JBZeldovic IDNovikov Struttura ed evoluzione delluniverso 1982[26] GCavalleri GSpinelli Found of Physics Vol13 No1212211983[27] MMamone Capria FPanbianco Found of Physics Vol24No68851994[28] SWHawking Physival Review D Vol37No49041988[29] PCaldirola Applicazioni e verifiche sperimentali della relativitagrave ristretta 1955[30] EWhittaker History of the theories of ether and electricity 1910[31] PAMDirac[32] AEinstein[33] see SMora Giornale di fisica 19[34] AKAMaciel JTiomno Foundations of Physics Vol19No55051989 [35] FWinterberg Zeitfur Naturforschung 43a11311988[36] DCMiller Nature Vol116No2096491925[37] GJoos Annalen der Physik73851930[38] WMHicks PhilMagS6Vol3No1391902[39] VSSony AmJPhysVol57No1211491989[40] Valentin Danci The Nineteen Postulates of Einsteins Special Relativity Theory httpswwwresearchgatenetpublication312553637_The_Nineteen_Postulates_of_Einstein27s_Special_Relativity_Theory[41] Eisenman MN Back to Galilean Transformation and Newtonian Physics Refuting the Theory of Relativity Journal of Physical Mathematics 2016 73[42] RSShankland ed altri Reviews of Modern Physics vol27n2p167aprile 1955 [43] MGSagnac Comptes Rendus 7081913 e 14101913[44] AAMichelson HGGale The Astrophysical Journal Vol61No31371925[45] MPLangevin Comptes Rendus 8311921[46] AALogunov YuVChugreev SovPhysUsp3198611988[47] WNernst Zeitfur PhysBd 1066331938[48] TSJaseia ed altri PhysRevVol133No5A12211964[49] ABrilletJLHall PhysRevLettVol42No95491979[50] HMinkovsky Gottinget Nachtrichten531908[51] see JCSlater Teoria quantistica della materia[52] compare to HEWilhelm Galileian Electrodynamics Vol1No5591990[53] WLShimmin GalilElectrVol5No3551994[54] HEWilhelm PhysEssaysVol6No34201993[55] RBDriscoll Hadronic JVol14No65011991[56] RLDavis KCHines AustJPhysVol45No55911992[57] FAMuller Foundof PhysLett Vol5No65911992

25

[58] SATolchelnikova-Murri GalilElectr Vol4No131993[59] EWSilvertooth PhysEssaysVol5No1821992[60] HCHayden GalilElectrVol2No3571991[61] DLBergman GalilElectrvol2No2301991[62] IF veda anche CWSherwin PhysRevAVol35No9 36501987[63] RLDavisEPSSheppard PhysRevLettVol63No19 20211989[64] Erwin Schroumldinger Quantisierung als Eigenwertproblem in Annalen der Physik vol 79 1926 pp 361-376 pp 489-527[65] alternative explanations BWSchumacher PhysEssaysVol6No44921993 oppure DJLarson PhysEssaysVol6No1661993[66] SLaplace Collected Papers Vedi anche RMonti Seagreen Nov1986[67] Hafele J C Keating R E (July 14 1972) Around-the-World Atomic Clocks Predicted Relativistic Time Gains (PDF) Science 177 (4044) 166ndash170 [68] Le nuove implicazioni dellrsquoEntanglement su Altrogiornaleorg URL consultato il 12 aprile 2016[69] Diamanti (quantisticamente) inseparabili in Le Scienze 2 dicembre 2011 URL consultato il 27 settembre 2014[70] Un nuovo stato della materia creato con lentanglement quantistico in Le Scienze 27 settembre 2014 URL consultato il 27 settembre 2014[71] Juan Yin Yuan Cao Yu-Huai Li ed altri Satellite-based entanglement distribution over 1200 kilometers in Science AAAS vol 356 nordm 6343 pp 1140 ndash 1144[72] Heacutector A Muacutenera Apeiron vol 5 No 1-2 (January-April 1998) 37-54[73] Hector A Munera Daniel Hernandez-Deckers German Arenas Edgar Alfonso and Ivan Lopez Observation of a Non-conventional Influence of Earthrsquos Motion on the Velocity of Photons and Calculation of the Velocity of Our Galaxy PIERS Proceedings Beijing China March 23ndash27 2009[74] Prochnow Dieter The Euclidean universendashAn alternative theory of relativity the General Science Journal[75] httpwwwetereedenergiadelvuotocomla_teoria_della_relativita_un_analisi_criticahtml[76] httpwwwanti-relativitycomHomehtm[77] httpwwwconservapediacomCounterexamples_to_Relativity[78] Catalogue of Errors for Both Theories of Relativity httpwwwkritik-relativitaetstheoriedeAnhaengeKapitel2-englischpdf[79] FSelleri WEAK RELATIVITY the physics of space and time without paradoxes[80] Ching-Chuan Su Modifications of the Lorentz Force Law Complying with Galilean Transformations and the Local-Ether Propagation Model[81] a similar but not coincident theory is developed here httpwwwneoclassicalrelativityorg[82] Musa D Abdullahi DEPENDENCE OF ACCELERATING FORCE ON VELOCITY OF A CHARGED PARTICLE MOVING IN AN ELECTRIC FIELD the General Science Journal

26

Index

1) Symbols

2) Special Relativity

3) Ether theory

31 - Michelson amp Morley

32 - Sagnac amp Michelson-Gale

33 ndash Cosmic Background Radiation

34 ndash More tests

4) Euclidean Space-time

41 ndash Length contraction

42 ndash Time dilation

43 ndash The γ factor

5) Electrodynamics

51 ndash Maxwell equations in invariant form

52 ndash Electromagnetic waves

53 ndash Lorentz Force


61 - Doppler effect and Ives amp Stilwell experiment

62 ndash Compton effect

64 - Fine structure of the Atom

66 - Hafele-Keating experiment

7) Schrodinger equation

2

1) Symbols





AF - absolute frame

IF - inertial frame



ET - ether theory





3










4



23) Ether theories





5


c2t 2=x2 (31)



(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)




2

(34)

that is

c = c-v c= - (c+v)


c = 1

radic ε 0 μ0(35)



6











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

1) Symbols





AF - absolute frame

IF - inertial frame



ET - ether theory





3










4



23) Ether theories





5


c2t 2=x2 (31)



(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)




2

(34)

that is

c = c-v c= - (c+v)


c = 1

radic ε 0 μ0(35)



6











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26










4



23) Ether theories





5


c2t 2=x2 (31)



(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)




2

(34)

that is

c = c-v c= - (c+v)


c = 1

radic ε 0 μ0(35)



6











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26



23) Ether theories





5


c2t 2=x2 (31)



(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)




2

(34)

that is

c = c-v c= - (c+v)


c = 1

radic ε 0 μ0(35)



6











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26


c2t 2=x2 (31)



(31) becomes

c2t 2=(x +v t )2=x 2 + v2 t 2+2 x v t (33)




2

(34)

that is

c = c-v c= - (c+v)


c = 1

radic ε 0 μ0(35)



6











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26











7






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26






34) More tests



8

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

34) Dynamics



) (41)


x =γ (x+β ct ) t=γ ( t+βx c) (42)


part2

c2 part t2 ϕ=∆ ϕ (43)




9


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26


gμν = gμν=iquest (1 0 0 00 1 0 00 0 1 00 0 0 1

) (44)





) (46)

So that



uμ=(c v ) aμ=(0a) (48)


f μ=m aμ (49)


pμ=( Ec

m v ) (410)

pμ pμ



m v2


10



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26



translate into

Λμν=(1 0 0 0β 1 0 00 0 1 00 0 0 1

) (413)




(415)


f μ=dpμ

dt=(0 a) (416)



minus vc

partpart x

part


c part t

partc partt

+ vc

partpart x

) (417)



) (418)


((1minusβ2) part2


part xpart

cpart tminus part2

part x2)ϕ=0 (419)


11






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26






12

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

42) Time dilation


t=lcm (424)




43) The γ factor





13

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

45) Electrodynamics





Aμ = (φ A)



A B = nabla times A



)(51)


f μ = (0 a) (52)


14


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26


ν Fμν =



) (53)




part pα

partt=q

cF

α β uβ

(55)









Maxwell Equations

partμ Fμρ=1c


15



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


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26



(510-12)

part Bpart t


part Dpart t







part

part x (minus2part

cpart tminus

part

part x)ϕ=0

(513)


partpartt

(minusβpart

c part tminus2

partpart x )ϕ=0

(514)


16



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


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25


26



part p α

partt=f α=q

cF

αβ u β


)(c+β ∙u

ux

uy

uz


f =qc





17









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26









18

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

follows

t1=Rc+t1 (61)

t2=t1+T+Rc


t2=t1+T+Rc







The result

=0 TT(1-vc)

= TT(1+vc)

=2 TT(1+v22c2)




m=(1+2)2=(1+v22c2sen2) (66)

=v22c2sen2

19





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26





0=hcsen-movsen

ho=h+moc2(1+2 -1)

from which





-o=(hmoc)(1-cos) (68)


T=ho-h (69)

because

20

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


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7 References


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25


26

=c=c(o+hmoc(1-cos))

resultsT=h





H0=p122m1+p2

22m2+V(|r1-r2|) (611)



The hamiltonian is

H=m1c2(1+12)+m2c2(1+2

2)+V(|r1-r2|) (612)


H=H0-(18)(m1v14c2+m2v2

4c2)

H=r2+V(r)-(18c2)(1-3(m1+m2))



21





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26





22



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26



(part2


(minus2 βpart

partxpart

c part tminus part2

part x2)ϕ=0

(71)




(ihpartpart t

minus h2

2m∆)ϕ=0

(72)




( part2



part x2)ϕ=0 (73)



( part2

c2part t 2 +m2 c4

h2 minus part2

part x2)ϕ=0

(74)


23

7 References


24


25


26

7 References


24


25


26


25


26


26

Theory of Events, Space and Time - · PDF fileTheory of Events, Space and Time a neoclassical...

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Transcript of Theory of Events, Space and Time - · PDF fileTheory of Events, Space and Time a neoclassical...